How many arrangements are there of the word MATHEMATICS?
Rule:
Start with the factorial of the number of letters in the word. Then,
for each indistinguishable letter in the word, divide by the factorial
of the number of times that letter occurs in the word.
"MATHEMATICS" is an 11-letter word. If all the letters were
distinguishable like in "MATHEmatICS", the answer would be 11! = 39916800
However, there are
2 indistinguishable M's
2 indistinguishable A's
2 indistinguishable T's
Thus, using the rule, we divide 11! by (2!)(2!)(2!)
\[\frac{ 11! }{ 2!*2!*2! }=\frac{ 39916800 }{ 8 }=4989600\]
How many of these start with the letter M?
That amounts to finding all the distinguishable arrangements
of the 10-letter "word" "ATHEMATICS" and putting an M in the beginning
of each.
"ATHEMATICS" is a 10-letter "word" and it contains
2 indistinguishable A's
2 indistinguishable T's
Thus, using the rule, we divide 10! by (2!)(2!)
\[\frac{ 10! }{ 2!*2! }=\frac{ 3628800 }{ 4 }=907200\]
How many of the arrangements in part a have the T’s together?
That amounts to finding all the distinguishable arrangements
of the 10-letter "word" "MATHEMAICS" and inserting another T to the right
of the "T" in each.
"MATHEMAICS" is a 10-letter "word" and it contains
2 indistinguishable M's
2 indistinguishable A's
Thus, using the rule, it's exactly the same answer as the second part.
We divide 10! by (2!)(2!)
\[\frac{ 10! }{ 2!*2! }=\frac{ 3628800 }{ 4 }=907200\]
There ya go!